Local description of closed submodules of a special module of entire functions of exponential type

Let $\pi_1(z),\dots,\pi_q(z)$ be a system of polynomials of the complex variable $z$. In connection with the problem of spectral synthesis for systems of differential operators $\pi_1(D),\dots,\pi_q(D)$, $D=d/dz$, the problem of the local description of closed submodules is considered for a special module of entire functions over the ring $\mathbb C[\pi_1,\dots,\pi_q]$. It is shown that this problem can be reduced to the local description over the ring $\mathbb C[l]$, where $l$ is the Luroth polynomial associated with the system $\pi_1(z),\dots,\pi_q(z)$.